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Random Walk Models for SpaceFractional Diffusion Processes
, 1998
"... . . . . . . . . . . . . . . . . . . . . . . . . p. 1 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . p. 2 2. The Standard Diffusion Equation . . . . . . . . . . . . . . p. 4 3. The Feller SpaceFractional Diffusion Equation . . . . . . . . p. 7 4. Random Walks for L'evyFeller Dif ..."
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Cited by 67 (12 self)
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. . . . . . . . . . . . . . . . . . . . . . . . p. 1 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . p. 2 2. The Standard Diffusion Equation . . . . . . . . . . . . . . p. 4 3. The Feller SpaceFractional Diffusion Equation . . . . . . . . p. 7 4. Random Walks for L'evyFeller Diffusion Processes . . . . . . p. 9 5. Proof of Convergence . . . . . . . . . . . . . . . . . . . p. 18 A. Fourier Transform and Pseudodifferential Operators . . . . . p. 20 B. The Stable Probability Distributions . . . . . . . . . . . . p. 21 Acknowledgements . . . . . . . . . . . . . . . . . . . . p. 26 References . . . . . . . . . . . . . . . . . . . . . . . p. 27 The paper is published in the new international journal Fractional Calculus & Applied Analysis, Vol. 1, No 2, pp. 167191, 1998. The topics of the journal, shortly referred to as FCAA J., are Fractional Calculus, Special Functions, Integral Transforms and closely related areas of Applied Analysis. For information on this new journal,...
Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 56 (14 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
Intensity process and compensator: A new filtration expansion approach and the Jeulin–Yor theorem. The Annals of Applied Probability
, 2007
"... Let (Xt)t≥0 be a continuoustime, timehomogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ ..."
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Cited by 15 (2 self)
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Let (Xt)t≥0 be a continuoustime, timehomogeneous strong Markov process with possible jumps and let τ be its first hitting time of a Borel subset of the state space. Suppose X is sampled at random times and suppose also that X has not hit the Borel set by time t. What is the intensity process of τ based on this information? This question from credit risk encompasses basic mathematical problems concerning the existence of an intensity process and filtration expansions, as well as some conceptual issues for credit risk. By revisiting and extending the famous Jeulin–Yor [Lecture Notes in Math. 649 (1978) 78–97] result regarding compensators under a general filtration expansion framework, a novel computation methodology for the intensity process of a stopping time is proposed. En route, an analogous characterization result for martingales of Jacod and Skorohod [Lecture Notes in Math. 1583 (1994) 21–35] under local jumping filtration is derived.
The Ants
, 1990
"... The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we procee ..."
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Cited by 7 (5 self)
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The paper deals with the numerical solution of the nonlinear Itô stochastic differential equations (SDEs) appearing in the unravelling of quantum master equations. We first develop an exponential scheme of weak order 1 for general globally Lipschitz SDEs governed by Brownian motions. Then, we proceed to study the numerical integration of a class of locally Lipschitz SDEs. More precisely, we adapt the exponential scheme obtained in the first part of the work to the characteristics of certain finitedimensional nonlinear stochastic Schrödinger equations. This yields a numerical method for the simulation of the mean value of quantum observables. We address the rate of convergence arising in this computation. Finally, an experiment with a representative quantum master equation illustrates the good performance of the new scheme. 1. Introduction. 1.1. Objectives. The primary objective of this paper is to develop an efficient
© 1999 PABST SCIENCE PUBLISHERS Confidence Intervals as an Alternative to Significance Testing
"... The article argues to replace null hypothesis significance testing by confidence intervals. Correctly interpreted, confidence intervals avoid the problems associated with null hypothesis statistical testing. Confidence intervals are formally valid, do not depend on apriori hypotheses and do not res ..."
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The article argues to replace null hypothesis significance testing by confidence intervals. Correctly interpreted, confidence intervals avoid the problems associated with null hypothesis statistical testing. Confidence intervals are formally valid, do not depend on apriori hypotheses and do not result in trivial knowledge. The first part presents critique of null hypothesis significance testing; the second part replies to critique against confidence intervals and tries to demonstrate their superiority to null hypothesis significance testing.
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"... Level sets of the stochastic wave equation driven by a symmetric Lévy noise∗ ..."
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Level sets of the stochastic wave equation driven by a symmetric Lévy noise∗
INVARIANT MEASURES ANID MARK0V CHAINS WITH RANDOM TRANSI'FION PROBABIILPTIES BY
"... Abstract. In this paper, suTTicient conditions for the existence of (afinite) invariant measures for a class of Markov chains with random transition probabilities are given. A special class of Markov chains with random transition probabilities is also studied here to show the relevance of attractor ..."
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Abstract. In this paper, suTTicient conditions for the existence of (afinite) invariant measures for a class of Markov chains with random transition probabilities are given. A special class of Markov chains with random transition probabilities is also studied here to show the relevance of attractors for certain iterated function systems to the invariant measures for these chains, and some of these results are illustrated with computergenerated pictures. I. Introduction. One of the aims of this paper is to generalize the model of Markov chains with random transition probabilities, extensively studied by Cogbum in a series of papers (here we mention only [ 5] ) and later by Orey [16], to a locally compact Hausdodl state space. We provide a sufficient condition for the existence of a nontrivial afinite and locally finite (i.e. finite on
A Measure Theoretic Approach to Information Retrieval
"... The vector space model of information retrieval is one of the classical and widely applied retrieval models. Paradoxically, it has been characterised by a discrepancy between its formal framework and implementable form. The underlying concepts of the vector space model are mathematical terms: linear ..."
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The vector space model of information retrieval is one of the classical and widely applied retrieval models. Paradoxically, it has been characterised by a discrepancy between its formal framework and implementable form. The underlying concepts of the vector space model are mathematical terms: linear space, vector, and inner product. However, in the vector space model, the mathematical meaning of these concepts is not preserved. They are used as mere computational constructs or metaphors. Thus, the vector space model actually does not follow logically from the mathematical concepts on which it has been claimed to rest. This problem has been recognised for more than two decades, but no proper solution has emerged so far. The present paper proposes just such a solution to this very problem. Firstly, the concept of retrieval is defined based on measure theory. Then, retrieval is particularised using fuzzy set theory. As a result, the retrieval function is conceived as the cardinality of the intersection of two fuzzy sets. This view makes it possible to build a connection to linear spaces. Thus, the classical and the generalised vector space models as well as the latent semantic indexing model gain a correct formal background with which they are consistent. At the same time it becomes clear that the inner product is not a necessary ingredient of the vector space model. Moreover, this view makes it possible to consistently formulate new retrieval methods: in linear space with general basis, entropybased, and probabilitybased. Experimental results using standard test collections are also reported.